Problem: $ D = \left[\begin{array}{rr}1 & 2 \\ 5 & 1 \\ -1 & 5\end{array}\right]$ $ E = \left[\begin{array}{rr}2 & -1 \\ 5 & -2\end{array}\right]$ What is $ D E$ ?
Explanation: Because $ D$ has dimensions $(3\times2)$ and $ E$ has dimensions $(2\times2)$ , the answer matrix will have dimensions $(3\times2)$ $ D E = \left[\begin{array}{rr}{1} & {2} \\ {5} & {1} \\ \color{gray}{-1} & \color{gray}{5}\end{array}\right] \left[\begin{array}{rr}{2} & \color{#DF0030}{-1} \\ {5} & \color{#DF0030}{-2}\end{array}\right] = \left[\begin{array}{rr}? & ? \\ ? & ? \\ ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ D$ , with the corresponding elements in column $j$ of the second matrix, $ E$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ D$ with the first element in ${\text{column }1}$ of $ E$ , then multiply the second element in ${\text{row }1}$ of $ D$ with the second element in ${\text{column }1}$ of $ E$ , and so on. Add the products together. $ \left[\begin{array}{rr}{1}\cdot{2}+{2}\cdot{5} & ? \\ ? & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ D$ with the corresponding elements in ${\text{column }1}$ of $ E$ and add the products together. $ \left[\begin{array}{rr}{1}\cdot{2}+{2}\cdot{5} & ? \\ {5}\cdot{2}+{1}\cdot{5} & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ D$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ E$ and add the products together. $ \left[\begin{array}{rr}{1}\cdot{2}+{2}\cdot{5} & {1}\cdot\color{#DF0030}{-1}+{2}\cdot\color{#DF0030}{-2} \\ {5}\cdot{2}+{1}\cdot{5} & ? \\ ? & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rr}{1}\cdot{2}+{2}\cdot{5} & {1}\cdot\color{#DF0030}{-1}+{2}\cdot\color{#DF0030}{-2} \\ {5}\cdot{2}+{1}\cdot{5} & {5}\cdot\color{#DF0030}{-1}+{1}\cdot\color{#DF0030}{-2} \\ \color{gray}{-1}\cdot{2}+\color{gray}{5}\cdot{5} & \color{gray}{-1}\cdot\color{#DF0030}{-1}+\color{gray}{5}\cdot\color{#DF0030}{-2}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rr}12 & -5 \\ 15 & -7 \\ 23 & -9\end{array}\right] $